16 CHAPTER 2. ADIABATIC THEORY
j1j 0, and from the solution Eq. (2.31), we see that this is equivalent to the criterion
jv1j cs, which is often used in the literature. For this specic solution, the perturbation
criterion demands
vbc
cs
��
2
: (2.32)
The loss factor �� and the quality factor Q of the system is related by Q = 1=(2��), so in
terms of Q the perturbation criteria Eq. (2.32) becomes
vbc
cs
4Q
: (2.33)
This sets a maximum amplitude of the wall oscillations for which the perturbation expansion
is valid. While this theoretical parallel plates example has a high Q 105, determined
only by bulk damping, typical water-lled rectangular microchannels driven at 2 MHz has
Q 400 24, 29, which is determined mainly by dissipation within the acoustic boundary
layers, discussed in Section 2.4.
2.3.2 Second-order time-averaged elds
The time-averaged second-order equations for the one-dimensional parallel plates system
becomes
1(��i!v1)
+ 0
v1@yv1
= ��c2s
@y
2
+ @2
y
v2
+
1
3 + b
@2
y
v2
; (2.34a)
0 = ��0@y
v2
�� @y
1v1
: (2.34b)
Inserting Eqs. (2.34b) and (2.24b) in Eq. (2.34a), and integrating Eq. (2.34b) with the
boundary conditions Eq. (2.22), yields
@y
2
=
2k2
0
!
1(iv1)
��
2��
!
@2
y
1v1
; (2.35a)
v2
= ��
1
0
1v1
: (2.35b)
Following the rule Eq. (2.16) for the time-average of products of rst-order elds, we obtain
the following solutions for v2 and @y2,
@y
2
=
k2
0
!
Re
��
1
iv1
��
��
!
@2
y Re
��
1
v1
; (2.36a)
v2
= ��
1
20
Re
��
1
v1
: (2.36b)
Inserting the rst-order solutions Eq. (2.28), the source term Re
��
1
v1
becomes
Re
��
1
v1
= ��0
v2b
c
cs
Re
i(1 �� i��)
sin(ky) cos(ky)
cos(k w
2 ) cos(k w
2 )
; (2.37)